So the rules, he proceeds to give in this way you saw them written out. This is a lovely chart that a wonderful historian of science, Eric Aiton, A I T O N, includes on his book, he wrote an entire book on the vortex theories. All the vortex theories of planetary motion.

I'll have his paper from the Cambridge History as a supplementary reading next week from this. But he prepared this chart, it's a nice chart to do things. We got in, effect seven cases. Let's start at the top. We've got two bodies with exactly the same, and it's his choice of the word him.

Rather the letter M. People just can't stop thinking that the notion of mass is so straightforward that Newton didn't have to invent it out of whole cloth. But he did. So was being used beforehand. Nobody used mass the way Newton used mass. And it was a struggle for Newton to use it, they come to it.

At any rate, they have the same quantity of matter in them, and they're going toward one another at equal speed. And remember no other bodies are around, these are the ideal case, where we do have the equivalent of a vacuum. Nothing else is touching these. And the answer is, they rebound In same speeds, in the opposite direction.

Now, there are two things to comment. You can see why that is. Neither can win the contest right? So they both have to respond equally. We have to assume De Carte had never played billiards. Right. Because what happens? They stop.

Yeah, that's right. They just stop. They come together like that.

Well, they can bounce off one another, but the case he's describing, it's a complex action to bounce off of one another. Second case, one of the bodies is bigger than the other, so it wins the contest, and the two go off together attached to one another. And that by the way, the solution on the right, that is algebraically the Huygens's solution, but that's a little bit more tricky than it sounds.

The third case is the same size, but one of the velocities is greater than the other. It wins the contest and they go off together. Those are the three. All three of those, they were both moving. Rules four, five and six has a body at rest. And now there's a funny set of claims.

A smaller body strikes a larger body at rest, and no matter what its speed is, it can't make the other body deviate from rest. I assume he'd seen, he was a military person, I assume he had seen bullets fired ,and I assume he'd seen the outcome. That's the conclusion and he, of course is gonna have an answer.

That's because that case I just described, doesn't have the object totally in isolation. We'll get to that. That's the point Corey was jumping on. The other case though, if the larger one strikes a body at rest, then they move off together. And if they're equal, and one of them has speed, and the other is at rest, you get a compromise.

One goes off at three-quarters of the speed, and the other goes off at a quarter of the speed. And then I'm not gonna go through the last three cases. What happens with the mixed cases, two different speeds and two different masses, two different magnitudes of the balls? And you get different combinations here.

And they're really compilations, combinations of the things that go before. Now the complaint, and the complaint ,was made by several people at the time, that nothing Descartes says in the Principia justifies these conclusions. They're arbitrary. So when one of his principle followers, asked him what's going on here, it's a long letter.

I'm not gonna read the whole thing, but I've given it to you. The principle is minimal mutation. It's a principle of minimum mutation. The least change, is the one that nature will resolve to, the least total change. And that actually gives you everything but six, it does not give you six.

The compromise in six, is not the least change. So there's a question whether Descartes had any principle basis that works all the way through these, that gives you his results. What happened historically is most people quit caring. Or what they started asking themselves, is what are the rules for billiard balls, when they strike one another?

And we're gonna see a series of solutions of that. The earliest one is by Huygens. Also, the most beautiful. Subsequently, by John Wallace, Christopher Wren, Newton, Mariotte. All sorts of people do it. Are 20th century solutions for what Newton offers in the Principia, for what it's worth. But that's neither here nor there.

It became a celebrated problem. What's most interesting about it in another respect is for Descartes, motion coming in spheres is a relatively rare thing. Most shapes of things in motion are not sphere, they're oddball shapes. Yet, the problem became one of what's the response of spheres? Except for the young Newton.

The young Newton decided, from day one, that he had to to do it for arbitrary shapes. And we'll read that paper the first time we. It was a very young paper. But he actually took Descartes seriously, that the problem was motion of arbitrary shapes, what happens on impact when arbitrary shapes hit one another, and he had sensible ideas, but he couldn't solve the problem.

The problem was a very difficult problem. In fact, the problem gets solved by Euler much much later. At any rate, those are the rules, and their importance to us is, that they created problems for other people.